Regions I, II and III are bounded by squares. The perimeter of region I is 12 units and the perimeter of region II is 24 units. What is the ratio of the area of region I to the area of region III? Express your answer as a common fraction.

[asy]
draw((0,0)--(9,0)--(9,9)--(0,9)--(0,0)--cycle,linewidth(2));
draw((9,0)--(15,0)--(15,6)--(9,6)--(9,0)--cycle,linewidth(2));
draw((9,6)--(12,6)--(12,9)--(9,9)--cycle,linewidth(2));
label("III",(4.5,4.5));
label("II",(12,3));
label("I",(10.5,7.5));
[/asy]
Answer: A side of square I has length 3, while a side of square II has length 6 (all sides have equal length).  Therefore, a side of square III has length 9.  Since the side length of square I is $\frac{1}{3}$ that of square III, and the ratio of their areas is the square of the ratio of their side lengths, the ratio of the area of square I to square III is $\left(\frac{1}{3}\right)^2 = \frac{1}{9}$.   Alternately, you can just calculate the areas: square I has an area of 9, square III has an area of 81, thus, the ratio of their areas is $\boxed{\frac{1}{9}}$